Curves in Banach spaces which allow a $C^2$ parametrization

Jakub Duda; Ludek Zajicek

doi:10.1112/jlms/jdq100

Curves in Banach spaces which allow a $C^2$ parametrization

Classical Analysis and ODEs 2014-02-26 v2 Differential Geometry

Authors: Jakub Duda , Ludek Zajicek

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Abstract

We give a complete characterization of those $f: [0,1] \to X$ (where $X$ is a Banach space which admits an equivalent Fr\'echet smooth norm) which allow an equivalent $C^2$ parametrization. For $X=\R$ , a characterization is well-known. However, even in the case $X=\R^2$ , several quite new ideas are needed. Moreover, the very close case of parametrizations with a bounded second derivative is solved.

Keywords

banach spaces banach space properties algebraic curves

Cite

@article{arxiv.math/0603735,
  title  = {Curves in Banach spaces which allow a $C^2$ parametrization},
  author = {Jakub Duda and Ludek Zajicek},
  journal= {arXiv preprint arXiv:math/0603735},
  year   = {2014}
}

Comments

22 pages; We split the original paper into two parts. This is the first part ($C^2$ paths), the second part (paths with finite convexity) will appear later

Curves in Banach spaces which allow a $C^2$ parametrization

Abstract

Keywords

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